9 METRIZABLE GROUPS 39 9. METRIZABLE GROUPS If G is a group, a function / on G x G is said to be left (resp. right) invariant if f(xy, xz) =/0, z) (resp. f(yx, zx) =f(y, z)) for all x, y, z in G. When G is commutative, these two conditions coincide, and/is then said to be translation-invariant. A distance d on G is left (resp. right) invariant if and only if the left (resp. right) translations are isometrics with respect to d. If / is a left-invariant function on G x G, then the function (x, y')^-^f(x"19 y"1) is right-invariant, and vice versa. For example, if E is a normed space, the distance ||jc — >>|| on E is transla- tion-invariant. (1 2.9.1 ) In order that the topology of a topological group G should be metriz- able (in which case G is said to be a metrizable group) it is necessary and sufficient that there should exist a denumerable fundamental system of neigh- borhoods of the neutral element #, whose intersection consists of e alone. When this condition is satisfied, the topology of G can be defined by a left- invariant distance , or by a right-invariant distance. It is enough to show that if there exists a denumerable fundamental system of neighborhoods (Un) of e in G such that Q Un = {e}, then the topology of G n can be defined by a left-invariant distance. We define inductively a sequence (Vn) of symmetric neighborhoods of e in G such that Vj c: V1 and for all n k 1 (12.8.3), so that (Vn) is also a fundamental system of neigh- borhoods of e. Now define a real-valued function g on G x G as follows : g(x, x) = 0; if x *£ y, then either x" 1y $ Vx, in which case we take g(x, y) = 1 ; or else there exists a greatest integer k such that x~1y e Vk (because x~ly ^ e cannot belong to all the Vn), in which case we define g(x, y) = 2"~k. It is clear from this definition that g(x, y) = g(y, x), that g(x, y) ;> 0, and that g(zx, zy) = g(x, y) for all x, y, z in G. Now let (12.9.1.1) «i(x,y)=in where the infimum is taken over the set of all finite sequences (z0 , zi9 . . . , zp) (with p arbitrary) such that ZQ = x and zp = y. We shall show that d is a left- invariant distance on G and satisfies the inequalities (12.9.12) tg(x, y) ^ d(x9 y) ^ g(x, y).roup so defined is called the product of the topological groups G« ,